The product of \( 2\pi \) and \( \frac{3}{4} \) is given by:
\[ 2\pi \times \frac{3}{4} = \frac{3}{2}\pi \]
To prove that \( \frac{3}{2}\pi \) is irrational, we note that \( \pi \) itself is an irrational number, and the product of a non-zero rational number (which \( \frac{3}{2} \) is) and an irrational number is always irrational. Thus, \( \frac{3}{2}\pi \) is irrational.